Eulerian and Lagrangian velocity statistics in compressible turbulence on a free surface

The statistics of velocity differences are analyzed for compressible turbulence on a free surface in both Eulerian and Lagrangian frames. Despite non-applicability of Kolmogorov 1941 theory (K41), prior measurements (J. Cressman et. al., New J. Phys., 6, 2004), have shown Kolmogorov scaling for Eulerian structure functions ($S_n(r) = \langle (\delta v_{||}(r))^n \rangle \sim r^{n/3}$). Here we measure the Eulerian third-order ($S_3(r)$) and Lagrangian second-order ($D_2(\tau)$) structure functions. The Eulerian third-moment is suprisingly consistent with K41 ($S_3(r) = -\frac{4}{5}\overline \varepsilon r)$). K41 predicts the lagrangian second-moment should scale linearly with time ($D_2(\tau) = C_0 \overline\varepsilon \tau$) in the inertial time-scales, however this scaling is not observed. Instead, the Lagrangian second-moment scales as $D_2(\tau) \sim \tau^{1/2}$. Absence of a suitable theory makes it difficult to explain the experimental observations.